Abstract

In this paper, we introduce the concept of q-set-valued α-quasi-contraction mapping and establish the existence of a fixed point theorem for this mapping in b-metric spaces. Our results are generalizations and extensions of the result of Aydi et al. (Fixed Point Theory Appl. 2012:88, 2012) and some recent results. We also state some illustrative examples to claim that our results properly generalize some results in the literature. Further, by applying the main results, we investigate a fixed point theorem in a b-metric space endowed with an arbitrary binary relation. At the end of this paper, we give open problems for further investigation.

MSC:47H10, 54H25.

Keywords

1 Introduction

Fixed point theory is one of the cornerstones in the development of mathematics since it plays a basic role in applications of many branches of mathematics. The famous Banach contraction principle is one of the most efficient power tools to study in this field since it can be observed easily and comfortably. In 1989, Bakhtin [1] introduced the concept of b-metric space and presented the contraction mapping in b-metric spaces that is generalization of the Banach contraction principle in metric spaces (see also Czerwik [2]). Subsequently, several researchers studied fixed point theory or the variational principle for single-valued and set-valued mappings in b-metric spaces (see [3–9] and references therein).

Recently, Aydi et al. [10] established the q-set-valued quasi-contraction mapping which is a generalization of the q-set-valued quasi-contraction mapping due to Amini-Harandi [11] in 2011. They also established a fixed point theorem for such a mapping in b-metric spaces. This theorem extends, unifies and generalizes several well-known comparable results in the existing literature.

On the other hand, Samet et al. [12] introduced the concept of α-admissible mapping and using this concept proved a fixed point theorem for a single-valued mapping. They also showed that these results can be utilized to derive fixed point theorems in partially ordered spaces and coupled fixed point theorems. Moreover, they applied the main results to ordinary differential equations. Recently, Mohammadi et al. [13] introduced the concept of α-admissible for a set-valued mapping which is different from the notion of α∗-admissible which was provided in [14]. Subsequently, there are a number of results via the concept of α-admissible mapping in many spaces (see [15–20] and references therein).

Inspired and motivated by Aydi et al. [10] and Mohammadi et al. [13], we introduce the class of q-set-valued α-quasi-contraction mappings and give a fixed point theorem for such mappings via the idea of α-admissible mapping. Our result improves and complements the main result of Aydi et al. [10] and many results in the literature. We also provide some examples to show the generality of our result. The applications for fixed point theorems in a b-metric space endowed with an arbitrary binary relation are also derived from our results. Furthermore, at the end of this paper, we give open problems for further investigation.

2 Auxiliary notions

Throughout this paper, the standard notations and terminologies in nonlinear analysis are used. For the convenience of the reader, we recall some of them. In the sequel, ℝ, R+ and ℕ denote the set of real numbers, the set of nonnegative real numbers and the set of positive integers, respectively.

Let X be a nonempty set, T:X→2X, where 2X is a collection of subsets of X and α:X×X→[0,∞). We say that

(1)

T is α∗-admissible if

for x,y∈X for which α(x,y)≥1⟹α∗(Tx,Ty)≥1,

where α∗(Tx,Ty):=inf{α(a,b)∣a∈Tx,b∈Ty}.

(2)

T is α-admissible whenever for each x∈X and y∈Tx with α(x,y)≥1, we have α(y,z)≥1 for all z∈Ty.

Remark 2.20 It is easy to prove that the set-valued mapping T is α∗-admissible implies that T is α-admissible mapping.

3 The existence of fixed point theorems for a set-valued mapping in b-metric spaces

In this section, we introduce the q-set-valued α-quasi-contraction mapping and obtain the existence of a fixed point theorem for such a mapping in b-metric spaces.

Definition 3.1 Let (X,d) be a b-metric space and α:X×X→[0,∞) be a mapping. The set-valued mapping T:X→Pb,cl(X) is said to be a q-set-valued α-quasi-contraction if

α(x,y)H(Tx,Ty)≤qM(x,y),

(3.1)

for all x,y∈X, where 0≤q<1 and

M(x,y)=max{d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)}.

Next, we give the main result in this paper.

Theorem 3.2Let(X,d)be a completeb-metric space (with constants≥1) such that theb-metric is a continuous functional onX×X, letα:X×X→[0,∞)be a mapping, and letT:X→Pb,cl(X)be aq-set-valuedα-quasi-contraction. Suppose that the following conditions hold:

Since q<1s2+s, we get qs2<1. From (3.6), we have d(u,Tu)=0. By Lemma 2.16, we get u∈Tu, that is, u is a fixed point of T. This completes the proof. □

Theorem 3.3Let(X,d)be a completeb-metric space (with constants≥1) such that theb-metric is a continuous functional onX×X, letα:X×X→[0,∞)be a mapping, and letT:X→Pb,cl(X)be aq-set-valuedα-quasi-contraction. Suppose that the following conditions hold:

Let(X,d)be a completeb-metric space (with constants≥1) such that theb-metric is a continuous functional onX×X, and letT:X→Pb,cl(X)be aq-set-valued quasi-contraction. If we setq<1s2+s, thenThas a fixed point inX, that is, there existsu∈Xsuch thatu∈Tu.

Remark 3.5 The condition of q in Theorem 3.2 becomes q<12 if we take s=1 (it corresponds to the case of metric spaces). Therefore, Theorems 3.2 and 3.3 are generalization of many known results in metric spaces.

The following example shows that Theorem 3.2 properly generalizes the main result, Theorem 2.2, of Aydi et al. [10].

Example 3.7 Let X=R with the functional d:X×X→R+ be defined by d(x,y):=|x−y|2. Clearly, (X,d) is a complete b-metric space with constant s=2. Define the set-valued mapping T:X→Pb,cl(X) by

Tx={[x,max{x,5}],x>1,[0,x4],0≤x≤1,[min{x,−5},x],x<0,

and α:X×X→[0,∞) by

α(x,y)={1,x,y∈[0,1],0,otherwise.

We obtain that H(T0,T4)=16 and M(0,4)=16. Therefore,

H(T0,T4)>qM(0,4)

for all 0≤q<1. This implies that the contraction condition of Theorem 2.2 of Aydi et al. [10] is not true for this case. Therefore, Theorem 2.2 cannot be used to claim the existence of a fixed point of T.

Next, we show that our result can be applied to this case. First of all, we show that T is a q-set-valued α-quasi-contraction mapping, where q=116. We need only to show the case of x,y∈[0,1] since the other case is trivial. For x,y∈[0,1], we have

α(x,y)H(Tx,Ty)=|x4−y4|2=|x−y|216=qd(x,y)≤qM(x,y).

This shows that T is a q-set-valued α-quasi-contraction mapping. Also, we have

q=116<16=1s2+s.

It is easy to check that T is an α-admissible mapping. For x0=1 and x1=0∈Tx0, we have α(x0,x1)≥1. Further, for any sequence {xn} in X with xn→x as n→∞ for some x∈X and α(xn,xn+1)≥1 for all n∈N, we obtain that α(xn,x)≥1 for all n∈N.

Therefore, all hypotheses of Theorem 3.2 are satisfied, and so T has infinitely many fixed points.

Next, we give the result for a single-valued mapping which is an extension of Corollary 2.4 of Aydi et al. [10] and the result of Ćirić [24].

Corollary 3.8Let(X,d)be a completeb-metric space (with constants≥1) such that theb-metric is a continuous functional onX×X, letα:X×X→[0,∞)be a mapping, and lett:X→Xbe aq-single-valuedα-quasi-contraction, that is,

If we setq<1s2+s, thenthas a fixed point inX, that is, there existsu∈Xsuch thatu=tu.

Proof It follows by applying Theorem 3.2 or 3.3. □

4 Applications on a b-metric space endowed with an arbitrary binary relation

In this section, we give the existence of fixed point theorems on a b-metric space endowed with an arbitrary binary relation.

Before presenting our results, we give the following definitions.

Definition 4.1 Let (X,d) be a b-metric space and ℛ be a binary relation over X. We say that T:X→Pb,cl(X) is a preserving mapping if for each x∈X and y∈Tx with xRy, we have yRz for all z∈Ty.

Definition 4.2 Let (X,d) be a b-metric space and ℛ be a binary relation over X. The set-valued mapping T:X→Pb,cl(X) is said to be a q-set-valued quasi-contraction with respect to ℛ if

H(Tx,Ty)≤qM(x,y)

(4.1)

for all x,y∈X for which xRy, where 0≤q<1 and

M(x,y)=max{d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)}.

Theorem 4.3Let(X,d)be a completeb-metric space (with constants≥1) such that theb-metric is a continuous functional onX×X, ℛbe a binary relation overXandT:X→Pb,cl(X)be aq-set-valued quasi-contraction with respect toℛ. Suppose that the following conditions hold:

If we setq<1s2+s, thenThas a fixed point inX, that is, there existsu∈Xsuch thatu∈Tu.

Proof Consider the mapping α:X×X→[0,∞) defined by

α(x,y)={1if xRy,0otherwise.

(4.2)

From condition (ii), we get α(x0,x1)≥1. It follows from T is a preserving mapping that T is an α-admissible mapping. Since T is a q-set-valued quasi-contraction with respect to ℛ, we have, for all x,y∈X,

α(x,y)H(Tx,Ty)≤qM(x,y).

(4.3)

This implies that T is a q-set-valued α-quasi-contraction mapping. Now all the hypotheses of Theorem 3.2 are satisfied and so the existence of the fixed point of T follows from Theorem 3.2. □

Next, we give some special case of Theorem 4.3 in partially ordered b-metric spaces. Before studying next results, we give the following definitions.

Definition 4.4 Let X be a nonempty set. Then (X,d,⪯) is called a partially ordered b-metric space if (X,d) is a b-metric space and (X,⪯) is a partially ordered space.

Definition 4.5 Let (X,d,⪯) be a partially ordered b-metric space. We say that T:X→Pb,cl(X) is a preserving mapping with ⪯ if for each x∈X and y∈Tx with x⪯y, we have y⪯z for all z∈Ty.

Definition 4.6 Let (X,d,⪯) be a partially ordered b-metric space. The set-valued mapping T:X→Pb,cl(X) is said to be a q-set-valued quasi-contraction with respect to ⪯ if

H(Tx,Ty)≤qM(x,y),

(4.4)

for all x,y∈X for which x⪯y, where 0≤q<1 and

M(x,y)=max{d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)}.

Corollary 4.7Let(X,d,⪯)be a complete partially orderedb-metric space (with constants≥1) such that theb-metric is a continuous functional onX×XandT:X→Pb,cl(X)be aq-set-valued quasi-contraction with respect to⪯. Suppose that the following conditions hold:

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